# How does a heated constant volume flow behave, for an expanding flow tube leading into the free atmosphere

I have a helium gas flow with a flow rate that has been set to a constant value ($0.3\,m^3/h$). It's streaming upwards in the picture, and beyond the upper big red dot position, it meets the atmospheric gas in the laboratory. The room has $p_{lab}=1\,atm$ and $T_{lab}=300\,K$. The gas has been heated up, so that when it reaches the lower small red dot position from below, it has to $800$ kelvins.

At the lower red dot position, the diameter switches from $D=3\,mm$ to $D=13\,mm$ and the tube that follows is $d_z=25\,mm$ long.

My question is how total gas pressure $p_g(z)$, gas temperature $T_g(z)$ and gas density $n(z)$ behaves along the $z$-axis, in particular inside the tube.

As I said, at the lower red dot $z=0$, we have $T_g(0)=800\,K$, and we may assume ideal gas conditions, $p=n\,kT$. I'm not primarily interested in the gas velocity. Instead of a sudden expansion point at the lower red dot, I'm fine with a suitable diameter profile smooth and simple $D(z)$, as long as $D(0)=3\,mm$ and $D(25mm)=13\,mm$. Albeit this will in generally result in a temperature distribution $T(\rho,z)$, where $\rho$ is a radial length, I'm interested in a mean value.

• I'm not sure that you will get an answer without experimental measurements or simulations. None of the lengths are long enough for a "fully developed" type flow to develop where you can make the usual assumptions to derive an analytical expression. You'll get a plume at the first diameter change, which may or may not spread wide enough to fill the larger tube which will again become a plume into the atmosphere. It may or may not be turbulent, you'll have recirculation vortices in the corners at the first expansion which will be air for awhile until it mixes in enough helium. – tpg2114 Feb 27 '15 at 19:01
• So... pretty interesting question but I'm not sure this is something we can answer here. And I've searched for papers with similar setups and can't seem to find any. – tpg2114 Feb 27 '15 at 19:02
• You might be able to use some ideas/theories that calculate centerline plume temperatures above a fire in the atmosphere, for example equation 1: fire.nist.gov/bfrlpubs/fire00/PDF/f00010.pdf Although you might want to go back to how expressions like that were developed and see what approximations/assumptions were used and whether they apply to you. – tpg2114 Feb 27 '15 at 19:04
• @tpg2114: Thanks for the response. Something more: Do you know rules of thumb to guess the temperature drop due to heat loss through the metal wall? (That might also depend on the metal in some way, I don't know how.) – Nikolaj-K Mar 2 '15 at 12:33
• @NikolajK I would just solve the heat equation in either 1D (if your gas is relatively uniform in temperature in the streamwise direction) or 2D (assume axisymmetric coordinates) if your gas is changing temperature in the streamwise direction. The only other thing you need to know is the jump conditions, usually just $[\lambda \partial T/\partial x]_{gas} = [\lambda \partial T/\partial x]_{metal}$. – tpg2114 Mar 2 '15 at 13:16